FPLO: Full-Potential Local-Orbital Approach to the

H. Eschrig, K. Koepernik, I. Opahle, and Manuel Richter

Leibniz-Institute for Solid State and Materials Research, IFW Dresden

FPLO: Full-Potential Local-Orbital Approach to the

Electronic Structure of Solids and Molecules

1. Introduction: Why yet another DFT solver?

2. Linear combination of local orbitals (LCLO) equations and core-valencetransformation

3. Optimum local basis: strategies

4. Performance and application

5. Summary and licence

Psi-k Training Graduate School

Bristol, March 25 to March 31, 2007

Dresden: founded in 1206; flooded in 2002

500.000 inhabitants, 7.000+ in microelectronics (AMD, Qimonda)

50.000 at University, 3 Max-Planck-Institutes, 3 Leibniz-Institutes,

several (10?) Fraunhofer-Institutes

BerlinHamburg

Munich Prague

D

IFW: Solid State and Materials Res.

≈ 400 employees, 65 staff scientists

ITF: Theoretical Solid State Physics

Group Numerical Solid State Physicsand Simulations with

8 senior scientists, 1 IT lady,3 postdocs, 3 PhD students,4 students, guests

≈ 200 CPUs


There are nine and sixty ways of constructing tribal lays,And-every-single-one-of-them-is-right.

(R. Kipling)

Headline to the chapter “Electronic states”

in Ziman’s textbook Principles of the Theory of Solids

Task:

Hψ = εψ single particle, e.g. Kohn − Sham

Existing solvers:

FLAPW/APW+lo, (FP-)LMTO, (screened) KKR, PP-PW,...


method advantages disadvantages

FLAPW/APW+lo accuracy only periodic structures,only ordered structures (no CPA)

LMTO, ASW speed, flexibility little accuracy,no high-lying states

FP-LMTO accuracy (?) only periodic structures

(screened) KKR flexibility (structures), performance vs. accuracyCPA, transport, high-ε states

PP-PW simplicity, restricted accuracy (heavy atoms),no Pulay forces only periodic, ordered structures

FPLO accuracy, speed, no high-ε statesflexibility (0D-3D), CPA


8.9 9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8

Lattice constant [a.u.]

−0.345

−0.344

To

tal en

erg

y +

26523 [

Hart

ree]

Thorium: total energy vs. lattice constant

FP−LMTO

LCGTO

FLAPW−6p1/2

FLAPWLMTO−ASA

exp.

RFPLO

State of the art, 2001:Lattice constants obtained bydifferent full-potential methodsdiffer by 3.5% (figure).

2007 (estimate): 0.5%

This corresponds to aconsiderably improved totalenergy stability, as

V = V0

(

1 − 1

B0

d∆Enum

dV+ . . .

)

Only with such a stability wecan address the improvementof DFT approximations.

Specified task: Kohn-Sham equations with periodic potential

(−∆

2+ Vext + VHartree + Vxc)ψkn(r) = ψkn(r)εkn

ρ(r) =

occ∑

kn

|ψkn(r)|2

VHartree(r) =

∫

d3r′ρ(r′)/|r − r′|

Vxc(r) = δExc[ρ]/δρ

“Kohn-Sham-equations”, a set of non-linear integro-differential equations, to besolved by iteration, intuition, and precise numerics.

“Total energy”: E[Vext], can be evaluated from ρ, VHartree, Vxc, and εkn.

“LDA”, “GGA”: particular approximations for Exc[ρ].

“LSDA”: ρ −→ ρss′ ; Vxc −→ Vxc + Bxc σ.

2. LCLO equations and core-valence transformation

Focus now on the solution of Hψ = εψ.

Blochs ansatz for his states ψkn(r),

ψkn(r) =1√N

∑

RsL

CLs,kn φsL(r − R − s) eik(R+s) .

Basis states φsL = 〈r|RsL〉: local overlapping orbitals at sites s in the cell at R

Quantum numbers: L = {ν, l, m}

Secular equation for a nonorthogonal basis:

∑

RsL

H |RsL〉 eik(R+s) CLs,kn =∑

RsL

|RsL〉 eik(R+s) CLs,kn ε


Solution of the secular equation:

∑

RsL

H |RsL〉 eik(R+s) CLs,kn =∑

RsL

|RsL〉 eik(R+s) CLs,kn ε

∑

RsL

〈0s′L′|H|RsL〉 eik(R+s−s

′) CLs,kn =∑

RsL

〈0s′L′|RsL〉 eik(R+s−s

′) CLs,kn ε

This yields the matrix eigenvalue problem

HC = SCε

with

Hs′L′,sL =∑

R

〈0s′L′|H|RsL〉 eik(R+s−s

′),

Ss′L′,sL =∑

R

〈0s′L′|RsL〉 eik(R+s−s

′).


To reduce the rank of the matrices, basis states are divided into the valencestates φsLv and the core states φsLc. The latter are defined by

〈R′s′L′

c|RsLc〉 = δcc′δRR′δss′

and can be removed from the basis by an exact transformation (A. Ernst, PhDthesis, TU Dresden, 1997). Let

S =

(

1 Scv

Svc Svv

)

=

(

1 0Svc SL

vv

) (

1 Scv

0 SRvv

)

= SLSR,

SLvvS

Rvv = Svv − SvcScv .

By definition of the core states,

H =

(

εc1 εcScv

Svcεc Hvv

)

, εc = diag(· · · , εsLc, · · · ).


Re-writing the secular equation,

(SL)−1H(SR)−1(SRC) = (SRC) ε

leads to the reduced problem

HvvCvv = Cvv εv ,

Hvv = (SLvv)

−1(Hvv − SvcHccScv)(SRvv)

−1

C =

1 −Scv(SRvv)

−1Cvv

0 (SRvv)

−1Cvv

.

This exact transformation allows to keep all electrons in the calculation, whilethe effort is reduced to the solution of the valence state problem.Problem: incompatibility of φν and φc in the scalar relativistic approximation.

3. Optimum local basis: strategies to define the φsL

The ancestor: LCAO

Linear Combination of Atomic Orbitals (Bloch, Slater & Koster, Callaway, ...)

The descendants:

Optimized LCAO (Dresden code, H. Eschrig, 1975-2000)

Optimized minimum local basis, implemented in FPLO-2 (2002) ... FPLO-5(2005)

Adjusted fixed local basis, implemented in DMol (B. Delley, PSI Villingen) andin FPLO-6 (K. Koepernik, IFW Dresden, 2007)

3. Optimum local basis: strategies to define the φsL

Solve the Kohn-Sham equations for a spherical atom (ion) and obtain thedensity:

ρ0(r) =

N∑

i=1

|φi(r)|2, φi : radial orbital functions

N = Z : atom, N < Z : ion.

Spherical atom (ion) Kohn-Sham potential:

V [ρ0] = Vext + VHartree + Vxc

= −Z

r+

∫

d3r′ρ0(r

′)

|r − r′| + Vxc(ρ0(r))

Linear combination of atomic orbitals (Callaway)

core states

semi-core states

valence states

r

E, V

V [ρ0]

conduct

ion

ban

ds

vale

nce

and energy window of incompleteness of LCAO

Optimized minimum basis (OLCAO ... FPLO5)

V0(r) = V [ρ0] +

(

r

ri

)n

, ri : optimization parameter, n = 4 or 5.

sufficient completeness in the energy window of valence and conduction bandscore states

semi-core states

r

E, V

V [ρ0]

conduct

ion

ban

ds

vale

nce

and

valence states

Optimized minimum basis, example: Au

Positions of the local state energies:

-10 -8 -6 -4 -2 0 2 4 6 8 10 12Energy [eV]

0

1

2

3

4to

talD

OS

[eV

-1]

5d3/2 5d5/2 6s 6p1/2 6p3/2

Optimized minimum local basis

The price of the minimum basis is that the basis orbitals must be well adjustedto the self-consistent potential. Therefore, in a calculation for a solid the basismust be recalculated in every self-consistency cycle.

The chance of this strategy is that an automatic basis optimization (ri) can beincorporated into the self-consistency cycle by applying a Hellmann-Feynmantype approach. (The total energy is optimized with respect to the (ri).)

This way, a very high accuracy of total energies and densities is achieved (inmost cases comparable with FLAPW-WIEN2k). For the first time ever, twoindependent all-electron schemes provided the same total energies,∆E/E ≈ 10−6 . . . 10−7, for a number of close-packed structures.

However, the optimization is not suited for the calculations of forces.

Adjusted fixed basis (FPLO6)

V0(r) = V [ρ0] +

(

r

α0rmax

)n0

+ Θ(rmax − r), α0 = 0.7 . . . 0.8, n0 = 14 . . . 18

rmax = . . . 8 . . . 12 a0

core states

semi-core states

r

E, V

V [ρ0]

conduct

ion

ban

ds

vale

nce

and

rat

rmax

V (r) = +∞ for r > rmax

V (r) = V [ρ0] for r < 1.5rat

V smooth


(Construction approved in the package DMol; courtesy of Bernhard Delley)

Self-consistent calculation of neutral quasi-atomic states and potential:

φi, ρ0 =

Z∑

i=1

|φi|2, V0(r) = V [ρ0] +

(

r

α0rmax

)n0

+ Θ(rmax − r)

Then, an ionic potential is constructed with N = Z − Q, Q = 0 . . . 6 ≤ Z,

ρ1 =

N∑

i=1

|φi|2, V1(r) = V [ρ1] +

(

r

α1rmax

)n1

+ Θ(rmax − r)

α1 and n1 may be chosen slightly smaller than α0 and n0.


core states

semi-core states

r

E, V

conduct

ion

ban

ds

vale

nce

and

rat

rmax

V [ρ1]

V [ρ1] − V [ρ0]

V [ρ0]


Construction of the fixed basis (numerical on an inhomogeneous radial grid) byminimization of dimer and close-package bonding energies with respect to Q.

Vk → Hk, Hkφi = φiεi

occupied, no overlap core states: φνl from V0

occupied, overlap semi-core states: φνl from V0

partly occupied valence states: φνl from V0

empty doubled valence state: φν+1,l from V1

empty polarization states: φνl from V1

core semi-core valence polarization rank of matrix

Al: (1s) (2s,2p) (3s,4s,3p,4p) (3d) 17Fe: (1s,2s,2p) (3s,3p) (3d,4d,4s,5s) (4p) 19

Example: Radial basis functions for Fe

0 2 4 6 8radius r in a.u.

-0.5

0

0.5

1

r ba

sis

func

tion

atomic orbitaloptimized basisdouble adapted fixed basis

3d valence basis of Fe

Dependence of total energies on the finite support radius

1 2 3 4 5dimer distance

-250

-200

-150

-100

-50

tota

l ene

rgy

[mH

a/pe

r at

om]

rmax= 8 aB

rmax=10 aB

rmax=14 aB

Fe dimerdependence on finite support radius

The total energies are shifted bya constant. Thus, a moderatecompression of the atomic stateshas no influence on forces and oncalculated geometries.

Dependence of total energies on the optimization charges

0 1 2 3 4 5 6 7 8 9positive excess charge Q

-240

-238

-236

-234

tota

l ene

rgy

[mH

a/pe

r at

om] 5s

4p4d

Fe dimercore : 3s3p / 4s3d 5s4p4d

1 mHa

0 1 2 3 4 5 6 7 8 9positive excess charge

-320

-319

-318

-317

tota

l ene

rgy

[mH

a/pe

r at

om]

5s4p4d

Fe bcccore: 3s3p / 4s3d 5s4p4d

1 mHa

Dimers and close packed systems can be treated with the same basis set.

4. Performance

method basis functions per atom remark

PP-PW > 500 considerable reduction by PAW

LAPW > 100

APW+lo > 50

FPLO 10 . . . 40 interpretation in chemical terms

LMTO 9 . . . 16 multiple basis for better accuracy

Note that the large-N scaling is with the third power of the matrix rank.

FPLO: no parallelization, except on shell level.

Performance check was carried out at my 5 years old laptop in the train fromLondon to Bristol.

0 2 4 6 8 10 12 14

number of atoms per u.c.

0

100

200

300ti

me

pe

r it

era

tio

n [

se

c]

Performance of FPLO−6

single processor laptop from 2002

scalar relativistic, Al supercells,

tetragonal symmetry,

number of k points indicated

3610950

760

24X24X(24/N)

570

380

182

N2

36X36X(36/N)

273

4. Performance: why do we need so many k points?

Converge the quantity you wish to calculate with respect to all parameters of thek-space integration.

If you once do this, you will avoid smearing techniques for metallic systemsforever.

Tetrahedron method: only one convergence parameter.

Examples will be given in the hands-on:

The spin moment of Ni changes by 5%, if you enhance the k-mesh density from123 to 243 points in the full Brillouin zone.

The DOS of Al changes (at some energies) by 10%, if you enhance the k-meshdensity from 243 to 483 points in the full Brillouin zone.

YCo5 - Balls and Spaghetti

Γ M K Γ A L H A

YCo5

−0.40

−0.20

0.00

0.20

0.40

Energ

y

ε n(k

) [eV

]

↑↓

Ferromagnetic ground state

YCo5 - pressure dependent DOS

A strong 1-D van-Hove-singularity is shiftedthrough EF.

The DOS shift isfacilitated by aconcomitant reductionof the moment andthe exchange split.

The large DOSyields a negativecompressibility.

YCo5 - Isostructural Phase Transition

∆V ≈ 1.6% at pc ≈ 20 GPaH. Rosner et al., Nature Physics, 2006

5. Summary http://www.FPLO.de/

- numerical atomic-like basis set, all-electron

- efficient: 10 (H) . . . 40 (Actinides) local functions per atom

- relative stability 10−6 (...−8) eV/atom, absolute accuracy about 0.1 eV/atom

- open core, LSDA+U (Igor Chaplygin, 2000)

- more features: (CPA; DLM); fat bands; Fermi surfaces and velocities; forces

- 4-component Dirac-Kohn-Sham, LSDA+OP (Ingo Opahle, 2001)

- cluster-version on the same footing as the periodic version

- 0D . . . 3D periodicity (slabs and rods in test, Ferenc Tasnadi)

- spin spirals with continuous variation of spin direction (Wenxu Zhang, in prep.)

- handy user interface, good portability, 105 lines source code, 20 person years

Advertisements and licence

http://www.FPLO.de/

- annual hands-on workshops (Daresbury, March 25-31, 2007)

- annual user- and developer meetings (La Plata, November 12-17, 2007)

Licence:

- cover fee (400 Euro) for academic use

- source code is provided and can be modified, but not distributed

- citation requested

- mailing list (do not hesitate to read the 50 manual pages!)

Acknowledgments

Permanent technical assistance by Ulrike Nitzsche.

Contributions by Igor Chaplygin (LSDA+U), Ferenc Tasnadi (Slab and rodversions in preparation), and Alim Ormeci (GGA).

Critical application, testing, completion, and distribution by Helge Rosner.

Funding: DFG, EC (psi-k f-electron).

Documents

FPLO: Full-Potential Local-Orbital Approach to the